Zeros of the second derivative of the modular $j$-function

Question

I am interested in the zeros of $j''(z)$, where $j:\mathbb{H}\rightarrow\mathbb{C}$ is the classic modular function. Specifically I am interested in knowing if the zeros of $j''$ are algebraic over $\mathbb{Q}$, or, even better, quadratic over $\mathbb{Q}$.